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Major Axis of Ellipse

Let ABC be ...

Question

Let $A B C$ be a triangle with in centre $I$ and in radius $r$ . Let $D, E, F$ be the feet of the perpendiculars from $I$ to the sides $B C, C A$ and $A B$ respectively. If $r_{1}, r_{2} r_{3}$ are the radii of circles inscribed in then prove $\frac{r_{1}}{r - r_{1}}_\frac{r_{2}}{r - r_{2}} + \frac{r_{3}}{r - r_{3}} = \frac{r_{1} r_{2} r_{3}}{(r - r_{1}) (r - r_{2}) (r - r_{3})}$

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Solution

$\frac{r - r_{1}}{r_{1}} = tan \frac{A}{2}$
$\frac{r_{1}}{r - r_{1}} = cot \frac{A}{2}$
$\sum cot \frac{A}{2} = cot \frac{A}{2} cot \frac{B}{2} cot \frac{C}{2}$
$= \frac{r_{1}}{r - r_{1}} \times \frac{r_{2}}{r - r_{2}} \times \frac{r_{3}}{r - r_{3}}$
$= \frac{r_{1} r_{2} r_{3}}{(r - r_{1}) (r - r_{2}) (r - r_{3})}$

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r_{1} + r_{2} + r_{3} - r = 4 R

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